When
k = = 0 the Friedmann
and fluid equations can readily
be solved for the equations of state given earlier, leading to the
classic cosmological solutions

(7)

(8)

In both cases the density falls as t-2. When k
= 0 we have the
freedom to rescale a and it is normally chosen to be unity at the
present, making physical and comoving scales coincide. The
proportionality constants are then fixed by setting the density to be
0 at time
t0, where here and throughout the subscript zero
indicates present value.

A more intriguing solution appears for domination by the cosmological
constant, namely

(9)

This is equivalent to the solution for a fluid with equation of
state p
= -.
The fluid equation then gives
= 0
and hence /
8G.

More complicated solutions can also be found for mixtures of
components. For example, if there is both matter and radiation the
Friedmann equation can be solved using conformal time
=
dt /
a, while if there is matter and a non-zero curvature term the
solution can be given either in parametric form using normal time t,
or in closed form with conformal time.